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Properties

Label 4046.2.a.f
Level $4046$
Weight $2$
Character orbit 4046.a
Self dual yes
Analytic conductor $32.307$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4046,2,Mod(1,4046)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4046.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4046, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4046 = 2 \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4046.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,2,1,0,-2,-1,-1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.3074726578\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} - q^{7} - q^{8} + q^{9} + 2 q^{12} - 4 q^{13} + q^{14} + q^{16} - q^{18} + 2 q^{19} - 2 q^{21} - 2 q^{24} - 5 q^{25} + 4 q^{26} - 4 q^{27} - q^{28} + 6 q^{29}+ \cdots - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 2.00000 1.00000 0 −2.00000 −1.00000 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( +1 \)
\(17\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4046.2.a.f 1
17.b even 2 1 14.2.a.a ✓ 1
51.c odd 2 1 126.2.a.b 1
68.d odd 2 1 112.2.a.c 1
85.c even 2 1 350.2.a.f 1
85.g odd 4 2 350.2.c.d 2
119.d odd 2 1 98.2.a.a 1
119.h odd 6 2 98.2.c.a 2
119.j even 6 2 98.2.c.b 2
136.e odd 2 1 448.2.a.a 1
136.h even 2 1 448.2.a.g 1
153.h even 6 2 1134.2.f.l 2
153.i odd 6 2 1134.2.f.f 2
187.b odd 2 1 1694.2.a.e 1
204.h even 2 1 1008.2.a.h 1
221.b even 2 1 2366.2.a.j 1
221.g odd 4 2 2366.2.d.b 2
255.h odd 2 1 3150.2.a.i 1
255.o even 4 2 3150.2.g.j 2
272.k odd 4 2 1792.2.b.g 2
272.r even 4 2 1792.2.b.c 2
323.c odd 2 1 5054.2.a.c 1
340.d odd 2 1 2800.2.a.g 1
340.r even 4 2 2800.2.g.h 2
357.c even 2 1 882.2.a.i 1
357.q odd 6 2 882.2.g.c 2
357.s even 6 2 882.2.g.d 2
391.c odd 2 1 7406.2.a.a 1
408.b odd 2 1 4032.2.a.w 1
408.h even 2 1 4032.2.a.r 1
476.e even 2 1 784.2.a.b 1
476.o odd 6 2 784.2.i.c 2
476.q even 6 2 784.2.i.i 2
595.b odd 2 1 2450.2.a.t 1
595.p even 4 2 2450.2.c.c 2
952.e odd 2 1 3136.2.a.e 1
952.k even 2 1 3136.2.a.z 1
1428.b odd 2 1 7056.2.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a ✓ 1 17.b even 2 1
98.2.a.a 1 119.d odd 2 1
98.2.c.a 2 119.h odd 6 2
98.2.c.b 2 119.j even 6 2
112.2.a.c 1 68.d odd 2 1
126.2.a.b 1 51.c odd 2 1
350.2.a.f 1 85.c even 2 1
350.2.c.d 2 85.g odd 4 2
448.2.a.a 1 136.e odd 2 1
448.2.a.g 1 136.h even 2 1
784.2.a.b 1 476.e even 2 1
784.2.i.c 2 476.o odd 6 2
784.2.i.i 2 476.q even 6 2
882.2.a.i 1 357.c even 2 1
882.2.g.c 2 357.q odd 6 2
882.2.g.d 2 357.s even 6 2
1008.2.a.h 1 204.h even 2 1
1134.2.f.f 2 153.i odd 6 2
1134.2.f.l 2 153.h even 6 2
1694.2.a.e 1 187.b odd 2 1
1792.2.b.c 2 272.r even 4 2
1792.2.b.g 2 272.k odd 4 2
2366.2.a.j 1 221.b even 2 1
2366.2.d.b 2 221.g odd 4 2
2450.2.a.t 1 595.b odd 2 1
2450.2.c.c 2 595.p even 4 2
2800.2.a.g 1 340.d odd 2 1
2800.2.g.h 2 340.r even 4 2
3136.2.a.e 1 952.e odd 2 1
3136.2.a.z 1 952.k even 2 1
3150.2.a.i 1 255.h odd 2 1
3150.2.g.j 2 255.o even 4 2
4032.2.a.r 1 408.h even 2 1
4032.2.a.w 1 408.b odd 2 1
4046.2.a.f 1 1.a even 1 1 trivial
5054.2.a.c 1 323.c odd 2 1
7056.2.a.bd 1 1428.b odd 2 1
7406.2.a.a 1 391.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4046))\):

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 6 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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